# Intersection of two functions

Could you please try to find the intersections points of this two functions: $$y = x$$ and $$y=(a+x) \ {\rm e}^ {-2(a+x)}$$ with $$x\geq0$$

To visualize the pairs of real numbers {x, a} that satisfy the equation,

you can use ContourPlot:

ContourPlot[(a + x) E^(-2 (a + x)) == x, {x, -2, 1}, {a, 0, 5}] or RegionPlot with options MeshFunctions + Mesh:

RegionPlot[True, {x, -2, 1}, {a, 0, 5},
BoundaryStyle -> None,
MeshFunctions -> {(#2 + #) E^(-2 (#2 + #)) - # &},
Mesh -> {{0}},
MeshStyle -> Directive[Thick, Red],
PlotStyle -> None,
PlotPoints -> 100] Looks like you'll need some constraints on a.

a = -1;
f[x_] := (a + x) E^(-2 (a + x))
Plot[{x, f[x]}, {x, 0.001, 4}, PlotRange -> {Automatic, {-2, 1}}] You can also use FindInstance. For example, for a given a:

FindInstance[{(a + x) Exp[-2 (a + x)] == x, a == 5}, {a, x}, Reals, 3] Or for a given range for a:

FindInstance[{(a + x) Exp[-2 (a + x)] == x, 0 < a < 5}, {a, x}, Reals, 10] Unfortunately, with Mathematica you can not get the solution, so I added the calculation with Maple here. Test:

eq /. {a -> 1, x -> 0.12}
0.12